# Full text of "Treatise On Analysis Vol-Ii"

## See other formats

```228       XIII   INTEGRATION

By (13.9.6) it is enough to show that (x, y)*-+f(x) is a v-measurable
mapping of X x Y into E. There exists a partition of X consisting of a
sequence (KJ of compact sets and a A-negligible set N such that the restriction
of/ to each Kn is continuous. For each compact subset L of Y, the restriction
of the mapping (x, y) H>/(JC) to each of the compact sets KM x L is continuous,
and the set N x Lis v-negligible (13.21.12), whence the result follows (13.9.4).

Given two mappings /: X -» R and #:Y-*R, we denote by f®g the
mapping (x, y) *->f(x)g(y) of X x Y into R (with the convention of (1 3.1 1 ) for
products in R). Similarly for mappings into C.

(13.21.14)   7/*/(resp. g) is a X-integrable (resp. n-integrable) mapping ofX
(resp. Y) into R or C, then the function f®g is v-integrable, and we have

(13.21.14.1)

By linearity we reduce to the case where /and g are mappings into R. By
virtue of (13.21.12), the set of points (x, y) E X x Y at which f(x) or g(y) is
infinite is v-negligible, hence it follows from (13.21.13) and (13.9.6) that/® g
is v-measurable. The fact that/® g is v-integrable then follows from (1 3.21 .11)
and (13.9.13). Finally, the formula (13.21.14.1) is a consequence of the Lebes-
gue-Fubini theorem.

(1 3.21 .1 5)   Let Abe a subset of X, and B a subset of Y. Then

(i)   v*(A x B) = l*(A),u*(B) (with the product convention of (1 3.1 1 )).

(ii)   If A is ^-measurable and B is ^-measurable, then A x B is v-
measurable.

(iii) If A is X-integrable and B is n~integrable9 then A x B is v-integrable,
and v(A x B) =

These assertions are particular cases of (13.21.11), (13.21.13) and
(13.21.14).

(13.21.16)   7jf/(resp. g) is a locally k-integrable mapping ofX (resp. locally
H-integrable mapping 0/Y) into R or C, thenf®g is locally v-integrable, and

(13.2116.1)             (f®g) • (1 ® u) = (/• A) ® (0 - 0).

Since the set of points (x, y) at which /(;c) or #G>) is infinite is v-negligible
(13.21.12), it follows from (13.21.13) and (13.9.8.1) that/® g is v-measurable.
Moreover, for each compact subset K (resp. L) of X (resp. Y) the function hand, the
```